Journal of Applied Electrochemistry

, Volume 37, Issue 5, pp 605–616 | Cite as

Adaptive, multi-parameter battery state estimator with optimized time-weighting factors

Original Paper


We derive and implement a battery control algorithm that can accommodate an arbitrary number of model parameters, with each model parameter having its own time-weighting factor, and we propose a method to determine optimal values for the time-weighting factors. Time-weighting factors are employed to give greater impact to recent data for the determination of a system’s state. We employ the (controls) methodology of weighted recursive least squares, and the time weighting corresponds to the exponential-forgetting formalism. The output from the adaptive algorithm is the battery state of charge (remaining energy), state of health (relative to the battery’s nominal performance), and predicted power capability. Results are presented for a high-power lithium ion battery.


Battery Control Equivalent circuit Mathematical model Power prediction State of charge prediction Weighted recursive least squares 

List of symbols


Coulombic capacity, C-h/s


1/C D , 1/F


Regressed intercept


1/ (R ct C D ), 1/s


Capacitance, F


Current, A


Number of parameters m to be regressed adaptively


Parameter to be regressed


Number of time steps (data points) in the regression


Power, W


Capacitance ratio,C D,discharge/C D,charge


High-frequency resistance, ohm


Effective interfacial resistance, ohm


Percent state of charge (energy content in the battery relative to the energy content upon full charge)


State of health, Eq. 15




Time, s


System voltage, V


Hysteresis voltage, V


Open-circuit voltage, V


Time-dependent values multiplying onto parameters m


Dependent variable


Weighting factor (Eq. 17)


Hysteresis parameter, C−1


Error or loss term


Unweighted total error as defined by Eq. 13


Forgetting factor (Eq. 2)


Parameter for selective weighting of data beyond that of the forgetting factor


Current efficiency





The authors recognize and thank Ramona Ying of General Motors R&D (Chemical and Environmental Sciences Laboratory) for acquiring the lithium ion battery data and helpful discussions with Damon Frisch and Brian Koch of the GM Electrical Center as well as Mutasim Salman of the Electrical Controls and Integration Laboratory at GM R&D.


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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.General Motors Research and DevelopmentWarrenUSA

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